Calculus Examples

$y = \frac{1}{3} x^{3} + x^{2} - 24 x + 15$

Step 1

Find the point at $x = - 3$ .

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Step 1.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{{(- 3)}^{3}}{3} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of ${(- 3)}^{3}$ and $3$ .

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Step 1.2.1.1.1

Rewrite $- 3$ as $- 1 (3)$ .

$f (- 3) = \frac{{(- 1 \cdot 3)}^{3}}{3} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.2

Apply the product rule to $- 1 (3)$ .

$f (- 3) = \frac{{(- 1)}^{3} \cdot 3^{3}}{3} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.3

Raise $- 1$ to the power of $3$ .

$f (- 3) = \frac{- 1 \cdot 3^{3}}{3} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.4

Factor $3$ out of $- 1 \cdot 3^{3}$ .

$f (- 3) = \frac{3 (- 1 \cdot 3^{2})}{3} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.5

Cancel the common factors.

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Step 1.2.1.1.5.1

Factor $3$ out of $3$ .

$f (- 3) = \frac{3 (- 1 \cdot 3^{2})}{3 (1)} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.5.2

Cancel the common factor.

$f (- 3) = \frac{3 (- 1 \cdot 3^{2})}{3 \cdot 1} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.5.3

Rewrite the expression.

$f (- 3) = \frac{- 1 \cdot 3^{2}}{1} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.1.5.4

Divide $- 1 \cdot 3^{2}$ by $1$ .

$f (- 3) = - 1 \cdot 3^{2} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.2

Rewrite $- 1 \cdot 3^{2}$ as $- 3^{2}$ .

$f (- 3) = - 3^{2} + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.3

Raise $3$ to the power of $2$ .

$f (- 3) = - 1 \cdot 9 + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.4

Multiply $- 1$ by $9$ .

$f (- 3) = - 9 + {(- 3)}^{2} - 24 \cdot - 3 + 15$

Step 1.2.1.5

Raise $- 3$ to the power of $2$ .

$f (- 3) = - 9 + 9 - 24 \cdot - 3 + 15$

Step 1.2.1.6

Multiply $- 24$ by $- 3$ .

$f (- 3) = - 9 + 9 + 72 + 15$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 9$ and $9$ .

$f (- 3) = 0 + 72 + 15$

Step 1.2.2.2

Add $0$ and $72$ .

$f (- 3) = 72 + 15$

Step 1.2.2.3

Add $72$ and $15$ .

$f (- 3) = 87$

Step 1.2.3

The final answer is $87$ .

$87$

Step 1.3

Convert $87$ to decimal.

$y = 87$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{3}}{3} + {(0)}^{2} - 24 \cdot 0 + 15$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{0}{3} + {(0)}^{2} - 24 \cdot 0 + 15$

Step 2.2.1.2

Divide $0$ by $3$ .

$f (0) = 0 + {(0)}^{2} - 24 \cdot 0 + 15$

Step 2.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 - 24 \cdot 0 + 15$

Step 2.2.1.4

Multiply $- 24$ by $0$ .

$f (0) = 0 + 0 + 0 + 15$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0 + 15$

Step 2.2.2.2

Add $0$ and $0$ .

$f (0) = 0 + 15$

Step 2.2.2.3

Add $0$ and $15$ .

$f (0) = 15$

Step 2.2.3

The final answer is $15$ .

$15$

Step 2.3

Convert $15$ to decimal.

$y = 15$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{{(- 2)}^{3}}{3} + {(- 2)}^{2} - 24 \cdot - 2 + 15$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 2$ to the power of $3$ .

$f (- 2) = \frac{- 8}{3} + {(- 2)}^{2} - 24 \cdot - 2 + 15$

Step 3.2.1.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{8}{3} + {(- 2)}^{2} - 24 \cdot - 2 + 15$

Step 3.2.1.3

Raise $- 2$ to the power of $2$ .

$f (- 2) = - \frac{8}{3} + 4 - 24 \cdot - 2 + 15$

Step 3.2.1.4

Multiply $- 24$ by $- 2$ .

$f (- 2) = - \frac{8}{3} + 4 + 48 + 15$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $4$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{8}{3} + \frac{4}{1} + 48 + 15$

Step 3.2.2.2

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4}{1} \cdot \frac{3}{3} + 48 + 15$

Step 3.2.2.3

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + 48 + 15$

Step 3.2.2.4

Write $48$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48}{1} + 15$

Step 3.2.2.5

Multiply $\frac{48}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48}{1} \cdot \frac{3}{3} + 15$

Step 3.2.2.6

Multiply $\frac{48}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48 \cdot 3}{3} + 15$

Step 3.2.2.7

Write $15$ as a fraction with denominator $1$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48 \cdot 3}{3} + \frac{15}{1}$

Step 3.2.2.8

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48 \cdot 3}{3} + \frac{15}{1} \cdot \frac{3}{3}$

Step 3.2.2.9

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (- 2) = - \frac{8}{3} + \frac{4 \cdot 3}{3} + \frac{48 \cdot 3}{3} + \frac{15 \cdot 3}{3}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 8 + 4 \cdot 3 + 48 \cdot 3 + 15 \cdot 3}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $4$ by $3$ .

$f (- 2) = \frac{- 8 + 12 + 48 \cdot 3 + 15 \cdot 3}{3}$

Step 3.2.4.2

Multiply $48$ by $3$ .

$f (- 2) = \frac{- 8 + 12 + 144 + 15 \cdot 3}{3}$

Step 3.2.4.3

Multiply $15$ by $3$ .

$f (- 2) = \frac{- 8 + 12 + 144 + 45}{3}$

Step 3.2.5

Simplify by adding numbers.

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Step 3.2.5.1

Add $- 8$ and $12$ .

$f (- 2) = \frac{4 + 144 + 45}{3}$

Step 3.2.5.2

Add $4$ and $144$ .

$f (- 2) = \frac{148 + 45}{3}$

Step 3.2.5.3

Add $148$ and $45$ .

$f (- 2) = \frac{193}{3}$

Step 3.2.6

The final answer is $\frac{193}{3}$ .

$\frac{193}{3}$

Step 3.3

Convert $\frac{193}{3}$ to decimal.

$y = 64.\overline{3}$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{{(- 1)}^{3}}{3} + {(- 1)}^{2} - 24 \cdot - 1 + 15$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{- 1}{3} + {(- 1)}^{2} - 24 \cdot - 1 + 15$

Step 4.2.1.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{1}{3} + {(- 1)}^{2} - 24 \cdot - 1 + 15$

Step 4.2.1.3

Raise $- 1$ to the power of $2$ .

$f (- 1) = - \frac{1}{3} + 1 - 24 \cdot - 1 + 15$

Step 4.2.1.4

Multiply $- 24$ by $- 1$ .

$f (- 1) = - \frac{1}{3} + 1 + 24 + 15$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $1$ as a fraction with denominator $1$ .

$f (- 1) = - \frac{1}{3} + \frac{1}{1} + 24 + 15$

Step 4.2.2.2

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{1}{1} \cdot \frac{3}{3} + 24 + 15$

Step 4.2.2.3

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + 24 + 15$

Step 4.2.2.4

Write $24$ as a fraction with denominator $1$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24}{1} + 15$

Step 4.2.2.5

Multiply $\frac{24}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24}{1} \cdot \frac{3}{3} + 15$

Step 4.2.2.6

Multiply $\frac{24}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24 \cdot 3}{3} + 15$

Step 4.2.2.7

Write $15$ as a fraction with denominator $1$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24 \cdot 3}{3} + \frac{15}{1}$

Step 4.2.2.8

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24 \cdot 3}{3} + \frac{15}{1} \cdot \frac{3}{3}$

Step 4.2.2.9

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{1}{3} + \frac{3}{3} + \frac{24 \cdot 3}{3} + \frac{15 \cdot 3}{3}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 + 3 + 24 \cdot 3 + 15 \cdot 3}{3}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $24$ by $3$ .

$f (- 1) = \frac{- 1 + 3 + 72 + 15 \cdot 3}{3}$

Step 4.2.4.2

Multiply $15$ by $3$ .

$f (- 1) = \frac{- 1 + 3 + 72 + 45}{3}$

Step 4.2.5

Simplify by adding numbers.

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Step 4.2.5.1

Add $- 1$ and $3$ .

$f (- 1) = \frac{2 + 72 + 45}{3}$

Step 4.2.5.2

Add $2$ and $72$ .

$f (- 1) = \frac{74 + 45}{3}$

Step 4.2.5.3

Add $74$ and $45$ .

$f (- 1) = \frac{119}{3}$

Step 4.2.6

The final answer is $\frac{119}{3}$ .

$\frac{119}{3}$

Step 4.3

Convert $\frac{119}{3}$ to decimal.

$y = 39.\overline{6}$

Step 5

Find the point at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{{(1)}^{3}}{3} + {(1)}^{2} - 24 \cdot 1 + 15$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

One to any power is one.

$f (1) = \frac{1}{3} + {(1)}^{2} - 24 \cdot 1 + 15$

Step 5.2.1.2

One to any power is one.

$f (1) = \frac{1}{3} + 1 - 24 \cdot 1 + 15$

Step 5.2.1.3

Multiply $- 24$ by $1$ .

$f (1) = \frac{1}{3} + 1 - 24 + 15$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Write $1$ as a fraction with denominator $1$ .

$f (1) = \frac{1}{3} + \frac{1}{1} - 24 + 15$

Step 5.2.2.2

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{1}{1} \cdot \frac{3}{3} - 24 + 15$

Step 5.2.2.3

Multiply $\frac{1}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{3}{3} - 24 + 15$

Step 5.2.2.4

Write $- 24$ as a fraction with denominator $1$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24}{1} + 15$

Step 5.2.2.5

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24}{1} \cdot \frac{3}{3} + 15$

Step 5.2.2.6

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24 \cdot 3}{3} + 15$

Step 5.2.2.7

Write $15$ as a fraction with denominator $1$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24 \cdot 3}{3} + \frac{15}{1}$

Step 5.2.2.8

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24 \cdot 3}{3} + \frac{15}{1} \cdot \frac{3}{3}$

Step 5.2.2.9

Multiply $\frac{15}{1}$ by $\frac{3}{3}$ .

$f (1) = \frac{1}{3} + \frac{3}{3} + \frac{- 24 \cdot 3}{3} + \frac{15 \cdot 3}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (1) = \frac{1 + 3 - 24 \cdot 3 + 15 \cdot 3}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Multiply $- 24$ by $3$ .

$f (1) = \frac{1 + 3 - 72 + 15 \cdot 3}{3}$

Step 5.2.4.2

Multiply $15$ by $3$ .

$f (1) = \frac{1 + 3 - 72 + 45}{3}$

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Add $1$ and $3$ .

$f (1) = \frac{4 - 72 + 45}{3}$

Step 5.2.5.2

Subtract $72$ from $4$ .

$f (1) = \frac{- 68 + 45}{3}$

Step 5.2.5.3

Add $- 68$ and $45$ .

$f (1) = \frac{- 23}{3}$

Step 5.2.5.4

Move the negative in front of the fraction.

$f (1) = - \frac{23}{3}$

Step 5.2.6

The final answer is $- \frac{23}{3}$ .

$- \frac{23}{3}$

Step 5.3

Convert $- \frac{23}{3}$ to decimal.

$y = - 7.\overline{6}$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 3 & 87 \\ - 2 & 64.333 \\ - 1 & 39.667 \\ 0 & 15 \\ 1 & - 7.667 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 3 & 87 \\ - 2 & 64.333 \\ - 1 & 39.667 \\ 0 & 15 \\ 1 & - 7.667 \end{array}$

Step 8